Or if you want, upgrade on your existing provider first, then setup streaming replication. But no matter what, pg_upgrade will require some form of downtime.
You could also use either Slony or londiste to directly migrate from Rackspace to an EC2 instance on 9.5; no need for DMS at all. That has the added benefit of allowing you to switch to a new version with virtually zero downtime (as in seconds if you script it) and allowing you to maintain the old version (along with any new data changes) in case you need to fail back (without data loss).
Keep in mind that in this case you're really only using DMS as a form of replication, so you might be better off just sticking with Postgres tools.
I'll be using DMS to migrate from an EC2 Instance to RDS instance. I won't be using EC2 on production environment, so to migrate to a RDS I need to use DMS...
I got some located servers at Rackspace, this means that I'm not using Rackspace postgres database service... so If i'd use slony I'd have to install/configure it on my own.
Thanks for the answers guys... This migration must happen and it will, and of course I'll keep u updated about how it went.